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Theorem cfss 10289
Description: There is a cofinal subset of 𝐴 of cardinality (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
cfss.1 𝐴 ∈ V
Assertion
Ref Expression
cfss (Lim 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem cfss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cfss.1 . . . . . 6 𝐴 ∈ V
21cflim3 10286 . . . . 5 (Lim 𝐴 → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
3 fvex 6910 . . . . . . 7 (card‘𝑥) ∈ V
43dfiin2 5037 . . . . . 6 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
5 cardon 9968 . . . . . . . . . 10 (card‘𝑥) ∈ On
6 eleq1 2817 . . . . . . . . . 10 (𝑦 = (card‘𝑥) → (𝑦 ∈ On ↔ (card‘𝑥) ∈ On))
75, 6mpbiri 258 . . . . . . . . 9 (𝑦 = (card‘𝑥) → 𝑦 ∈ On)
87rexlimivw 3148 . . . . . . . 8 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) → 𝑦 ∈ On)
98abssi 4065 . . . . . . 7 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ⊆ On
10 limuni 6430 . . . . . . . . . . . 12 (Lim 𝐴𝐴 = 𝐴)
1110eqcomd 2734 . . . . . . . . . . 11 (Lim 𝐴 𝐴 = 𝐴)
12 fveq2 6897 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
1312eqcomd 2734 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (card‘𝐴) = (card‘𝑥))
1413biantrud 531 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ( 𝐴 = 𝐴 ↔ ( 𝐴 = 𝐴 ∧ (card‘𝐴) = (card‘𝑥))))
15 unieq 4919 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴 𝑥 = 𝐴)
1615eqeq1d 2730 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → ( 𝑥 = 𝐴 𝐴 = 𝐴))
171pwid 4625 . . . . . . . . . . . . . . . . 17 𝐴 ∈ 𝒫 𝐴
18 eleq1 2817 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐴𝐴 ∈ 𝒫 𝐴))
1917, 18mpbiri 258 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴𝑥 ∈ 𝒫 𝐴)
2019biantrurd 532 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → ( 𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴)))
2116, 20bitr3d 281 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → ( 𝐴 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴)))
2221anbi1d 630 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → (( 𝐴 = 𝐴 ∧ (card‘𝐴) = (card‘𝑥)) ↔ ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥))))
2314, 22bitr2d 280 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)) ↔ 𝐴 = 𝐴))
241, 23spcev 3593 . . . . . . . . . . 11 ( 𝐴 = 𝐴 → ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
2511, 24syl 17 . . . . . . . . . 10 (Lim 𝐴 → ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
26 df-rex 3068 . . . . . . . . . . 11 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)))
27 rabid 3449 . . . . . . . . . . . . 13 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
2827anbi1i 623 . . . . . . . . . . . 12 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)) ↔ ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
2928exbii 1843 . . . . . . . . . . 11 (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
3026, 29bitri 275 . . . . . . . . . 10 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
3125, 30sylibr 233 . . . . . . . . 9 (Lim 𝐴 → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥))
32 fvex 6910 . . . . . . . . . 10 (card‘𝐴) ∈ V
33 eqeq1 2732 . . . . . . . . . . 11 (𝑦 = (card‘𝐴) → (𝑦 = (card‘𝑥) ↔ (card‘𝐴) = (card‘𝑥)))
3433rexbidv 3175 . . . . . . . . . 10 (𝑦 = (card‘𝐴) → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥)))
3532, 34spcev 3593 . . . . . . . . 9 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) → ∃𝑦𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥))
3631, 35syl 17 . . . . . . . 8 (Lim 𝐴 → ∃𝑦𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥))
37 abn0 4381 . . . . . . . 8 ({𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅ ↔ ∃𝑦𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥))
3836, 37sylibr 233 . . . . . . 7 (Lim 𝐴 → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅)
39 onint 7793 . . . . . . 7 (({𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ⊆ On ∧ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅) → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
409, 38, 39sylancr 586 . . . . . 6 (Lim 𝐴 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
414, 40eqeltrid 2833 . . . . 5 (Lim 𝐴 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
422, 41eqeltrd 2829 . . . 4 (Lim 𝐴 → (cf‘𝐴) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
43 fvex 6910 . . . . 5 (cf‘𝐴) ∈ V
44 eqeq1 2732 . . . . . 6 (𝑦 = (cf‘𝐴) → (𝑦 = (card‘𝑥) ↔ (cf‘𝐴) = (card‘𝑥)))
4544rexbidv 3175 . . . . 5 (𝑦 = (cf‘𝐴) → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥)))
4643, 45elab 3667 . . . 4 ((cf‘𝐴) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥))
4742, 46sylib 217 . . 3 (Lim 𝐴 → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥))
48 df-rex 3068 . . 3 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)))
4947, 48sylib 217 . 2 (Lim 𝐴 → ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)))
50 simprl 770 . . . . . . . 8 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴})
5150, 27sylib 217 . . . . . . 7 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
5251simpld 494 . . . . . 6 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ∈ 𝒫 𝐴)
5352elpwid 4612 . . . . 5 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥𝐴)
54 simpl 482 . . . . . . 7 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → Lim 𝐴)
55 vex 3475 . . . . . . . . . 10 𝑥 ∈ V
56 limord 6429 . . . . . . . . . . . 12 (Lim 𝐴 → Ord 𝐴)
57 ordsson 7785 . . . . . . . . . . . 12 (Ord 𝐴𝐴 ⊆ On)
5856, 57syl 17 . . . . . . . . . . 11 (Lim 𝐴𝐴 ⊆ On)
59 sstr 3988 . . . . . . . . . . 11 ((𝑥𝐴𝐴 ⊆ On) → 𝑥 ⊆ On)
6058, 59sylan2 592 . . . . . . . . . 10 ((𝑥𝐴 ∧ Lim 𝐴) → 𝑥 ⊆ On)
61 onssnum 10064 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑥 ⊆ On) → 𝑥 ∈ dom card)
6255, 60, 61sylancr 586 . . . . . . . . 9 ((𝑥𝐴 ∧ Lim 𝐴) → 𝑥 ∈ dom card)
63 cardid2 9977 . . . . . . . . 9 (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
6462, 63syl 17 . . . . . . . 8 ((𝑥𝐴 ∧ Lim 𝐴) → (card‘𝑥) ≈ 𝑥)
6564ensymd 9026 . . . . . . 7 ((𝑥𝐴 ∧ Lim 𝐴) → 𝑥 ≈ (card‘𝑥))
6653, 54, 65syl2anc 583 . . . . . 6 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ≈ (card‘𝑥))
67 simprr 772 . . . . . 6 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (cf‘𝐴) = (card‘𝑥))
6866, 67breqtrrd 5176 . . . . 5 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ≈ (cf‘𝐴))
6951simprd 495 . . . . 5 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 = 𝐴)
7053, 68, 693jca 1126 . . . 4 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))
7170ex 412 . . 3 (Lim 𝐴 → ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)) → (𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴)))
7271eximdv 1913 . 2 (Lim 𝐴 → (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)) → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴)))
7349, 72mpd 15 1 (Lim 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wex 1774  wcel 2099  {cab 2705  wne 2937  wrex 3067  {crab 3429  Vcvv 3471  wss 3947  c0 4323  𝒫 cpw 4603   cuni 4908   cint 4949   ciin 4997   class class class wbr 5148  dom cdm 5678  Ord word 6368  Oncon0 6369  Lim wlim 6370  cfv 6548  cen 8961  cardccrd 9959  cfccf 9961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-er 8725  df-en 8965  df-dom 8966  df-card 9963  df-cf 9965
This theorem is referenced by: (None)
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